Beyond Linear Response: Equivalence between Thermodynamic Geometry and Optimal Transport.

Area of Science:

• Thermodynamics
• Statistical Mechanics
• Non-equilibrium Systems

Background:

• Thermodynamic geometry provides a framework for understanding the geometry of thermodynamic state spaces.
• Optimal control protocols in the slow-driving limit are known to follow geodesics of the friction tensor.
• Extending these optimal protocols beyond the linear response regime remains a significant challenge.

Purpose of the Study:

• To demonstrate the equivalence between thermodynamic geometry and L^2 optimal transport geometry for overdamped dynamics.
• To develop a computationally tractable method for obtaining optimal protocols beyond the slow-driving limit.
• To explain observed phenomena in optimal protocols, such as nonmonotonic behavior and discontinuous jumps.

Main Methods:

• Equating thermodynamic geometry with L^2 optimal transport geometry for overdamped systems.
• Formulating optimal protocols as a sum of friction tensor geodesics and a counterdiabatic term.
• Utilizing the Fisher information metric to define the counterdiabatic term.

Main Results:

• Thermodynamic geometry is shown to be equivalent to L^2 optimal transport geometry for overdamped dynamics.
• A method is presented for computing optimal protocols beyond the slow-driving limit, involving geodesic paths and a counterdiabatic correction.
• The derived geodesic-counterdiabatic protocols accurately describe parametric harmonic potentials and reproduce complex behaviors in double-well systems.

Conclusions:

• The connection between thermodynamic geometry and optimal transport provides a powerful tool for analyzing non-equilibrium systems.
• The developed method offers a computationally feasible approach to finding optimal driving protocols in various regimes.
• This work elucidates the origins of nonmonotonicity and abrupt changes in optimal control protocols.